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Zbl 0645.34007
Staněk, Svatoslav
A phase of the differential equation $y''=Q(t)y$ with a complex coefficient Q of the real variable.
(English)
[J] Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 85, Math. 25, 57-75 (1986). ISSN 0231-9721

Let Q be a continuous complex-valued function on $j=(a,b)$ $(-D\le a<b\le \infty)$, Im Q(t)$\not\equiv 0$. Then there exist independent solutions $y\sb 1,y\sb 2$ of (Q): $y''=Q(t)y$ such that $y\sp 2\sb 1(t)+y\sp 2\sb 2(t)\ne 0$ for $t\in j$. We say that a $C\sp 3$ complex-valued function $\alpha$ is a (first) phase of (Q) if there exist independent solutions u, v of (Q), $u\sp 2(t)+v\sp 2(t)\ne 0$ on j, such that $\alpha '(t)=- w/(u\sp 2(t)+v\sp 2(t)),$ $t\in j$, where $w=uv'-u'v$. In the real case the phase of the second-order differential equation was introduced by {\it O. Boruvka} [Linear Differential Transformations of the Second Order (1971; Zbl 0222.34002)]. If $\alpha$ is a phase of (Q) then $b\sp{i\alpha (t)}/\sqrt{\alpha'(t)}$, $e\sp{-i\alpha (t)}/\sqrt{\alpha'(t)}$ are independent solutions of (Q). Phases of (Q) are used for the investigation of the decomposition of zeros of solutions to the differential equation (Q).
[S.Staněk]
MSC 2000:
*34A30 Linear ODE and systems
34M99 Differential equations in the complex domain

Keywords: phase; second-order differential equation; decomposition of zeros of solutions

Citations: Zbl 0222.34002

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